OP never mentioned whether the nanny accepted the job via phone or in person. She just mentions that the nanny sent several emails expressing her excitement about starting the job. But the latest email was a week before the anticipated start date, and then the nanny sent the final sorry-but-I've-decided-to-go-in-a-different-direction email. So I don't think declining by email is the worst thing or unprofessional behavior since their communication was primarily by email.
Furthermore, the nanny obviously knew that the OP would be upset by the news, but the nanny had already made her decision. So she probably didn't want to engage the OP further. Many times on this board, people post WWYD about difficult conversations when they know the other person will be upset by their decisions, and the overwhelming advice is to keep things short and not offer explanations/excuses that the other person will try to argue around and change your mind.
Who knows why the nanny decided to renege? Perhaps one of her glowing previous families was able to hire her back. Or perhaps one of them found a friend to hire her. If you were a nanny, you might wish to go with a family that is personally known by your previous families. However wonderful the OP is, a "known" quantity is a little safer in this type of arrangement; that's why as parents we prefer nannies that are referred by our friends as well.
I agree with the other PPs that there was no way that the nanny could have given OP the news that would have sat well with her. It is understandably a frustrating situation for the OP, but if the nanny found another position that she likes better for whatever reason--benefits, pay, known family, shorter commute, better hours, etc.--that's pretty much how it is. PP said that the nanny should have asked the OP to match whatever other offer she has, but perhaps that would have frustrated the OP as well. OP may have felt trapped and may have matched the offer to ensure childcare, but she may well have (understandably) resented the last-minute change in contract and then started looking for a new nanny. As a nanny, you might prefer to just take the better offer rather than risk upsetting your new employer from the start.
OP--I wish you the best as you search for a new nanny. Sorry that it turned out like this. I would be disappointed too.